Deformations of the central extension of the Poisson superalgebra

0.24 MB·English

by  S. E. Konstein

Deformations of the central extension of
the Poisson superalgebra.
S.E.Konstein∗ and I.V.Tyutin†‡
5
0 I.E.Tamm Department of Theoretical Physics,
0
P. N. Lebedev Physical Institute,
2

n 119991, Leninsky Prospect 53, Moscow, Russia.
a
J

5

1 Abstract
v
7
Poisson superalgebras realized on the smooth Grassmann valued functions with compact
2
0 support in Rn have the central extensions. The deformations of these central extensions are
1 found.
0
5
0
1 Introduction
/
h
t
-
p This paper continues the investigation started in [2], [3] and [4]. We consider the Poisson
e superalgebra realized on smooth Grassmann-valued functions with compact support in Rn.
h
: As it is shown in [2] this superalgebra has central extensions for some dimensions. For these
v
i dimensions, we found the second adjoint cohomology space and the deformations of the
X
Poisson superalgebra under consideration.
r
a
1.1 Poisson superalgebra
Let K be either R or C. We denote by D(Rn) the space of smooth K-valued functions with
compact support on Rn. This space is endowed with its standard topology. We set
Dn− = D(Rn+)⊗Gn−, En− = C∞(Rn+)⊗Gn−, D′n− = D′(Rn+)⊗Gn−,
n+ n+ n+
where Gn− is the Grassmann algebra with n generators and D′(Rn+) is the space of con-
−
tinuous linear functionals on D(Rn+). The generators of the Grassmann algebra (resp., the
coordinates of the space Rn+) are denoted by ξα, α = 1,...,n (resp., xi, i = 1,...,n ). We
− +
shall also use collective variables zA which are equal to xA for A = 1,...,n and are equal
+
to ξA−n+ for A = n +1,...,n +n . The spaces Dn−, En−, and D′n− possess a natural
+ + − n+ n+ n+
grading which is determined by that of the Grassmann algebra. The parity of an element
∗E-mail: [email protected]
†E-mail: [email protected]
‡ThisworkwassupportedbytheRFBR(grantsNo.02-02-16944(I.T.)andNo.02-02-17067(S.K.)), and
by the grant LSS-1578.2003.2.
1
2
f of these spaces is denoted by ε(f). We also set ε = 0 for A = 1,...,n and ε = 1 for
A + A
A = n +1,...,n +n . Each function f ∈ En− can be expressed in the form
+ + − n+
f(z) = f (x)ξα1...ξαn−.
α1,...,αn− 1 n−
αXi=0,1
We will use the notation supp(f) = supp(f ) ⊂ Rn+.
←− {αi} α1,...,αn−
Let ∂/∂zA and ∂ /∂zA be the operators of the left and right differentiation. The Poisson
S
bracket is defined by the relation
←−
∂ ∂
{f,g}(z) = f(z) ωAB g(z) = −(−1)ε(f)ε(g){g,f}(z), (1)
∂zA ∂zB
where the symplectic metric ωAB = −(−1)εAεBωBA is a constant invertible matrix. For
definiteness, we choose it in the form
ωij 0
ωAB = , λ = ±1, i,j = 1,...,n , α,β = 1+n ,...,n +n ,
0 λ δαβ α + + − +
α
(cid:18) (cid:19)
where ωij is the canonical symplectic form (if K = C, then one can choose λ = 1). The
α
nondegeneracy of the matrix ωAB implies, in particular, that n is even. The Poisson
+
superbracket satisfies the Jacobi identity
(−1)ε(f)ε(h){f,{g,h}}(z)+cycle(f,g,h) = 0, f,g,h ∈ En−. (2)
n+
By Poisson superalgebra, we mean the space Dn− with the Poisson bracket (1) on it. The
n+
relations (1) and (2) show that this bracket indeed determines a Lie superalgebra structure
on Dn−.
n+
The integral on Dn− is defined by the relation f¯d=ef dzf(z) = dx dξf(z), where
n+ Rn+
the integral on the Grassmann algebra is normed by the condition dξξ1...ξn− = 1. We
R R R
identify Gn− with its dual space G′n− setting f(g) = dξf(ξ)g(ξ), f,g ∈ Gn−. Correspond-
R
ingly, the space D′n− of continuous linear functionals on Dn− is identified with the space
n+ R n+
D′(Rn+) ⊗ Gn−. As a rule, the value m(f) of a functional m ∈ D′n− on a test function
n+
f ∈ Dn− will be written in the “integral” form: m(f) = dzm(z)f(z).
n+
R
1.2 Cohomologies of Poisson superalgebra
Let L be a Lie superalgebra acting in a Z -graded space V (the action of f ∈ L on v ∈ V
2
will be denoted by f ·v). The space C (L,V) of p-cochains consists of all multilinear super-
p
antisymmetric mappings from Lp to V. The space C (L,V) possesses a natural Z -grading:
p 2
by definition, M ∈ C (L,V) has the definite parity ε(M ) if
p p p
ε(M (f ,...,f )) = ε(M )+ε(f )+...+ε(f )
p 1 p p 1 1
3
for any f ∈ L with parities ǫ(f ). The differential dV is defined to be the linear operator
j j p
from C (L,V) to C (L,V) such that
p p+1
p+1
dVp Mp(f1,...,fp+1) = − (−1)j+ε(fj)|ε(f)|1,j−1+ε(fj)εMpfj ·Mp(f1,...,f˘j,...,fp+1)−
j=1
X
− (−1)j+ε(fj)|ε(f)|i+1,j−1M (f ,...f ,{f ,f },f ,...,f˘,...,f ), (3)
p 1 i−1 i j i+1 j p+1
i<j
X
for any M ∈ C (L,V) and f ,...f ∈ L having definite parities. Here the sign˘means
p p 1 p+1
that the argument is omitted and the notation
j
|ε(f)| = ε(f )
i,j l
l=i
X
has been used. The differential dV is nilpotent (see [1]), i.e., dV dV = 0 for any p =
p+1 p
0,1,.... The p-thcohomologyspace of the differential dV will bedenoted by Hp. The second
p V
cohomologyspaceH2 intheadjointrepresentationiscloselyrelatedtotheproblemoffinding
ad
formal deformations of the Lie bracket {·,·} of the form {f,g} = {f,g}+~2{f,g} +....
∗ 1
The condition that {·,·} is a 2-cocycle is equivalent to the Jacobi identity for {·,·} modulo
∗
the ~4-order terms.
In [2] and [4], we studied the cohomologies of the Poisson algebra Dn− in the following
n+
representations:
1. The trivial representation: V = K, f · a = 0 for any f ∈ Dn− and a ∈ K. The
n+
space C (Dn−,K) consists of separately continuous antisymmetric multilinear forms
on (Dn−p)p. n+The cohomology spaces and the differentials is denoted by Hp and dtr
n+ tr p
respectively.
2. The adjoint representation: V = Dn− and f ·g = {f,g} for any f,g ∈ Dn−. The space
n+ n+
C (Dn−,Dn−) consists of separately continuous antisymmetric multilinear mappings
p n+ n+
from (Dn−)p to Dn−. The cohomology spaces and the differentials is denoted by Hp
n+ n+ ad
and dad respectively.
p
The following theorems are proved in [2] and [4]:
Theorem 1
Let bilinear forms C1 and C2 be defined by the relations
2 2
C1(f,g) = f¯g¯, (4)
2
C2(f,g) = dz f(z)E g(z)−(−1)ε(f)ε(g)g(z)E f(z) , f,g ∈ Dn−, (5)
2 z z n+
Z
(cid:0) (cid:1)
where E d=ef 1− 1zA ∂ . 1
z 2 ∂zA
If n is even and n 6= n +4, then H2 = 0;
− − + tr
if n = n +4, then H2 ≃ K and the form C2 is a nontrivial cocycle;
− + tr 2
if n is odd, then H2 ≃ K and the form C1 is a nontrivial cocycle.
− tr 2
1The operator Ez is a derivation of the Poisson superalgebra.
4
Theorem 2 Let V be the one-dimensional subspace of C (Dn−,Dn−) generated by the
1 1 n+ n+
cocycle
m = E f(z).
1|2 z
Then there is a natural isomorphismV ⊕(En−/Dn−) ≃ H1 taking(M ,T) ∈ V ⊕(En−/Dn−)
1 n+ n+ ad 1 1 n+ n+
to the cohomology class determined by the cocycle M (z|f) + {t(z),f(z)}, where t ∈ En−
1 n+
belongs to the equivalence class T.
Theorem 3
Let n ≥ 4 or (n = 2 and n ≥ 4)
+ + −
Let V be the subspace of C (Dn−,Dn−) generated by the bilinear mappings m and m
2 2 n+ n+ 2|1 2|3
from (Dn−)2 to Dn−, which are defined by the relations
n+ n+
←− 3
∂ ∂
m (z|f,g) = f(z) ωAB g(z), (6)
2|1 ∂zA ∂zB
!
m (z|f,g) = g¯E f −(−1)ε(f)ε(g)f¯E g. (7)
2|3 z z
Then the cochains m and m are independent nontrivial cocycles, dimV = 2, and
2|1 2|3 2
there is a natural isomorphism V ⊕(En−/Dn−) ≃ H2 taking (M ,T) ∈ V ⊕(En−/Dn−) to
2 n+ n+ ad 2 2 n+ n+
the cohomology class determined by the cocycle
M (z|f,g)−{t(z),f(z)}g¯+(−1)ε(f)ε(g){t(z),g(z)}f¯,
2
where t ∈ En− belongs to the equivalence class T.
n+
Theorem 4
1. Let n = 2, 0 ≤ n ≤ 3. Let N (f) = −2Λ(x ) duθ(x −y )f(u), where Λ ∈ C∞(R)
+ − 1 2 1 1
be a function such that d Λ ∈ D(R) and Λ(−∞) = 0, Λ(+∞) = 1. Let
dx R
z = (x ,x ,ξ , ..., ξ ), u = (y ,y ,η , ..., η )
1 2 1 n− 1 2 1 n−
Θ(z|f) = duδ(x −y )θ(x −y )f(u),
1 1 2 2
Z
NE(z|f,g) = Θ(z|∂ fg)−Θ(z|f∂ g)−2(−1)n−ε(f)∂ f(z)Θ(z|g)+2Θ(z|f)∂ g(z),
2 2 2 2 2
ND(f,g) = NE(f,g)+dadN (f,g), (8)
2 2 1 1
∆(z|f) = duδ(x−y)f(u), (9)
Z
∆ (z|f) = duη δ(x−y)f(u). (10)
α α
Z
Then the bilinear mappings Ln− from (Dn−)2 to En− defined by the relations
2 2 2
1 1
L0(x|f,g) = ND(x|f,g)+ xi∂ f(x) g(x)− f(x) xi∂ g(x) ,
2 2 2 i 2 i
L1(z|f,g) = ND(z|f,g)−∆((cid:0)z|f)g(z)(cid:1)+(−1)ε(f)f(z)∆(cid:0)(z|g) (cid:1)
2 2
2
− (−1)ε(f) ξ1∂ f(z) ∆(z|g), (11)
ξ1
3
L2(z|f,g) = ND(z|f,g)−(cid:0) ∆(z|f)g((cid:1)z)+f(z)∆(z|g)
2 2
L3(z|f,g) = ND(z|f,g)−∆(z|f)g(z)+(−1)ε(f)f(z)∆(z|g)+
2 2
+∂ f(z)∆ (z|g)−(−1)ε(f)∆ (z|f)∂ g(z),
ξα α α ξα
5
are cocycles and maps (Dn−)2 to Dn−.
2 2
2. Let n = 2 and 0 ≤ n ≤ 3. Let Vn− be the subspace of C (Dn−,Dn−) generated by
+ − 2 2 2 2
the cocycles M1, M3 and Ln−, where the cocycles M1 and M3 are defined in Theorem
2 2 2 2 2
3.
Then there is a natural isomorphism Vn−⊕(En−/Dn−) ≃ H2 taking (M ,T) ∈ Vn−⊕
2 2 2 ad 2 2
(En−/Dn−) to the cohomology class determined by the cocycle
2 2
M (z|f,g)−{t(z),f(z)}g¯+(−1)ε(f)ε(g){t(z),g(z)}f¯,
2
where t ∈ En− belongs to the equivalence class T.
2
1.3 Central extensions
Let L be the central extension of the superalgebra L, i.e. L = L⊕c, f = f +aI ∈ L, aI ∈ c,
ε(I) = 0, a ∈ K. The bracket in L we denote as [f,g] = −(−1)ε(f)ε(g)[g,f],
[f,g] = {f,g}+C(f,g)I, C(f,g) ∈ K, [f,I] = 0,
where C(f,g) = −(−1)ε(f)ε(g)C(g,f), ε(C) = 0, is a generator of the second cohomology in
the trivial representation of the algebra L:
dtrC(f,g,h) = C({f,g},h)−(−1)ε(g)ε(h)C({f,h},g)−C(f,{g,h}) = 0.
2
It follows from Theorem 1, that the Poisson superalgebra Dn− has the central extension
n+
Dn− either if n is odd or if n = n +4.
n+ − − +
Consider C (L,L).
p
If M ∈ C (L,L) then M (f,...) = M (f,...) + m (f,...)I ∈ L, where M ∈ C (L,L),
p p p p p p p
m ∈ C (L,K), ε(M ) = ε(M ) = ε(m ) = ε .
p p p p p Mp
The differential dad is defined in a usual way.
For linear forms, the differential has the form:
dadM (f,g) = [M (f),g]−(−1)ε(f)ε(g)[M (g),f]−M ([f,g])
1 1 1 1 1
and can be expressed on the decomposition of L as
dadM (I,I) ≡ 0 (12)
1 1
dadM (f,I) = −{M (I),f}−C(M (I),f)I, (13)
1 1 1 1
dadM (f,g) = dadM (f,g)−M (I)C(f,g)+γ(f,g)I, (14)
1 1 1 1 1
where
γ(f,g) = C(M (f),g)−(−1)ε(f)ε(g)C(M (g),f)−m ({f,g})−m (I)C(f,g). (15)
1 1 1 1
For bilinear forms, the differential has the form:
dadM (f,g,h) = (−1)ε(g)ε(h)[M (f,h),g]−[M (f,g),h]−
2 2 2 2
−(−1)ε(f)ε(g)+ε(f)ε(h)[M (g,h),f]−
2
−M ([f,g],h)+ (−1)ε(g)ε(h)M ([f,h],g)+M (f,[g,h])
2 2 2
6
and can be expressed on the decomposition of L as
dadM (f,g,h) = dadM (f,g,h)+
2 2 2 2
+[M (f,I)C(g,h)−(−1)ε(f)ε(g)M (g,I)C(f,h)+(−1)ε(f)ε(h)+ε(g)ε(h)M (h,I)C(f,g)]+
2 2 2
+[(−1)ε(g)ε(h)C(M (f,h),g)−C(M (f,g),h)−(−1)ε(f)ε(g)+ε(f)ε(h)C(M (g,h),f)+
2 2 2
+m (f,I)C(g,h)−(−1)ε(f)ε(g)m (g,I)C(f,h)+(−1)ε(f)ε(h)+ε(g)ε(h)m (h,I)C(f,g)−
2 2 2
−dtrm (f,g,h)]I, (16)
2 2
dadM (f,g,I) = dadM˜ (f,g)+[C(M (f,I),g)−(−1)ε(f)ε(g)C(M (g,I),f)−
2 2 1 1 2 2
−m ({f,g},I)]I, (17)
2
dadM (f,I,I) = dadM (I,I,I) ≡ 0, (18)
2 2 2 2
where
M˜ (f) = M (f,I).
1 2
1.4 Cohomologies of central extension of the Poisson superalgebra
The following theorems are proved in the next sections.
Theorem 5
Let n = 2k+1. Let the bilinear mappings M1 and M3 from (Dn−)2 to Dn− are defined
− 2 2 n+ n+
by the relations
M1(z|f,g) = m (z|f,g), M1(f,I) = 0, (19)
2 2|1 2
M3(z|f,g) = m (z|f,g), M3(f,I) = 0, (20)
2 2|3 2
and the bilinear mapping L from (D1)2 to D1 is defined by the relations
2 2
20
L(z|f,g) = L1(z|f,g)+ dzξf(z)g¯− dzξg(z)f¯ I, L(f,I) = 0, (21)
2 3
(cid:18)Z Z (cid:19)
where m (z|f,g), m (z|f,g), and L1(z|f,g) are defined by (6), (7), and (11) respectively.
2|1 2|1 2
Let V be the subspace of C (Dn−,Dn−) generated by the bilinear mappings M1, M3, and
2 2 n+ n+ 2 2
δ δ L.
k,0 2,n+
Then dimV = 2+δ δ , and there is a natural isomorphism V ⊕(E2k+1/D2k+1) ≃ H2
2 k,0 2,n+ 2 n+ n+ ad
taking (M ,T) ∈ V ⊕(En−/Dn−) to the cohomology class determined by the cocycle
2 2 n+ n+
M (z|f,g)−{t(z),f(z)}g¯+(−1)ε(f)ε(g){t(z),g(z)}f¯,
2
where t ∈ En− belongs to the equivalence class T.
n+
Theorem 6
Let n = n +4. Let the bilinear mappings M3 and Q from (Dn−)2 to Dn− are defined
− + 2 τ n+ n+
by the relations
M3(z|f,g) = g¯E f −(−1)ε(f)ε(g)f¯E g, M3(f,I) = 0, (22)
2 z z 2
Q (z|f,g) = {τ(z),g(z)}f¯−(−1)ε(f)ε(g){τ(z),f(z)}g¯+ C2(τ,g)f¯−(−1)ε(f)ε(g)C2(τ,f)g¯ I
τ 2 2
Q (z|f,I) = 0, (23)
τ (cid:0) (cid:1)
7
where C2 is defined by (5) and ζ ∈ En−.
2 n+
Let V be the subspace of C (Dn−,Dn−) generated by the bilinear mapping M3.
2 2 n+ n+ 2
Then dimV = 1 and there is a natural isomorphism V ⊕(E2k+1/D2k+1) ≃ H2 taking
2 2 n+ n+ ad
(M ,T) ∈ V ⊕(En−/Dn−) to the cohomology class determined by the cocycle
2 2 n+ n+
M (z|f,g)+Q (z|f,g)
2 τ
where τ ∈ En− belongs to the equivalence class T.
n+
1.5 Deformations of the Lie superalgebra
LetLbeatopologicalLiesuperalgebra overKwithLiesuperbracket {·,·},K[[~2]]bethering
offormalpower seriesin~2 overK, andL[[~2]]betheK[[~2]]-moduleofformalpower series in
~2 with coefficients in L. We endow both K[[~2]] and L[[~2]] by the direct-product topology.
The grading of L naturally determines a grading ofL[[~2]]: an element f = f +~2f +...has
0 1
a definite parity ε(f) if ε(f) = ε(f ) for all j = 0,1,... Every p-linear separately continuous
j
mapping from Lp to L (in particular, the bracket {·,·}) is uniquely extended by K[[~2]]-
linearity to a p-linear separately continuous mapping over K[[~2]] from L[[~2]]p to L[[~2]]. A
(continuous) formal deformation of L is by definition a K[[~2]]-bilinear separately continuous
Lie superbracket C(·,·) on L[[~2]] such that C(f,g) = {f,g} mod ~2 for any f,g ∈ L[[~2]].
Obviously, every formal deformation C is expressible in the form
C(f,g) = {f,g}+~2C (f,g)+~4C (f,g)+..., f,g ∈ L, (24)
1 2
where C are separately continuous skew-symmetric bilinear mappings from L×L to L (2-
j
cochains with coefficients in the adjoint representation of L). Formal deformations C1 and
C2 are called equivalent if there is a continuous K[[~2]]-linear operator T : L[[~2]] → L[[~2]]
such that TC1(f,g) = C2(Tf,Tg), f,g ∈ L[[~2]].
Here as well as in [3], we use more restricting definition
Definition 7 Formal deformations C1 and C2 are called similar if there are a continuous
K[[~2]]-linear operators T, T : L[[~2]] → L[[~2]] such that TC1(f,g) = C2(Tf,Tg), f,g ∈
1
L[[~2]] and Tf = f +~2T f.
1
1.6 Deformations of central extension of the Poisson superalgebra
The following theorems are proved in subsequent sections:
←−
Theorem 8 Let n = 2k+1. Let M (z|f,g) = 1 f(z)sinh ~κ ∂ ωAB ∂ g(z). For given
− κ ~κ ∂zA ∂zB
κ, let ζ(z),w(z) ∈ E2k+1[[~2]] satisfy the following conditions(cid:16) (cid:17)
n+
i) ε(ζ) = 1, ε(w) = 0; (25)
ii) M (z|ζ,ζ)+w(z) ∈ D2k+1[[~2]]. (26)
κ n+
8
Let N (f,g) = N (z|f,g)+n (f,g)I, where
2|κ,ζ,w 2|κ,ζ,w 2|κ,ζ,w
N (z|f,g) = M (z|f +~2ζf¯,g +~2ζg¯)+~4w(z)f¯g¯,
2|κ,ζ,w κ
N (z|f,I) = M (z|w,f +~2ζf¯),
2|κ,ζ,w κ
¯
n (f,g) = fg¯,
2|κ,ζ,w
n (f,I) = 0,
2|κ,ζ,w
N (z|I,I) = 0.
2|κ,ζ,w
Then
1. every continuous formal deformation of the superalgebra D2k+1 is similar to the super-
n+
bracket
[f,g] = N (f,g)
κ,ζ,w 2|κ,ζ,w
with some κ and ζ(z),w(z) ∈ E2k+1[[~2]] satisfying (25)-(26) ;
n+
2. the superbracket [f,g] is similar to the superbracket [f,g] , if ζ (z) − ζ(z) ∈
κ,ζ1,w1 κ,ζ,w 1
Dn−[[~2]] and w (z)−w(z) ∈ Dn−[[~2]].
n+ 1 n+
Theorem 9 Let n = n + 4. Let ζ(z) ∈ En++4[[~2]], c ∈ K[[~2]], C(f,g) =
− + n+ 3
dz f(z)E g(z)−(−1)ε(f)ε(g)g(z)E f(z) and
z z
R (cid:0) S (f,g) = S (z|f,g)(cid:1)+s (f,g)I,
2|ζ,c3 2|ζ,c3 2|ζ,c3
S (z|f,g) = {f(z),g(z)}+c f¯E g(z)−(−1)ε(f)ε(g)g¯E f(z) +
2|ζ,c3 3 z z
+~2 {ζ(z),g(z)}f¯−(−1)ε(f)ε(g){ζ(z),f(z)}g¯ ,
(cid:0) (cid:1)
s (f,g) = C(f,g)+C(~2ζ,g)f¯−(−1)ε(f)ε(g)C(~2ζ,f)g¯,
2|ζ,c3 (cid:0) (cid:1)
S (f,I) = s (f,I) = 0.
2|ζ,c3 2|ζ,c3
Then
1. every continuous formal deformation of the superalgebra Dn++4 is similar to the su-
n+
perbracket
[f,g] = S (f,g)
ζ,c3 2|ζ,c3
with some c and ζ(z) ∈ E2k+1[[~2]] ;
3 n+
2. the superbracket [f,g] is similar to the superbracket [f,g] , if ζ (z) − ζ(z) ∈
ζ1,c3 ζ,c3 1
Dn−[[~2]].
n+
2 Superalgebra D2k+1
n
+
2.1 Second adjoint cohomology
In the superalgebra under consideration we have C(f,g) = f¯g¯ and [f,g] = {f,g}+f¯g¯I.
Consider the cohomology equation dadM (f,g,h) = 0. It follows from (17) that
2 2
dadM˜ (z|f,g) = 0 and so, due to Theorem 2
1 1
M (z|f,I) = t0E f(z)+{t′(z),f(z)}, t′(z) ∈ E2k+1, ε(t0) = ε(t′(z)) = ε , (27)
2 z n+ M2
9
It follows from (17) also that m ({f,g},I) = (2+n −n )t0f¯g¯ which implies t0 = 0 and
2 + −
m ({f,g},I) = 0. In its turn, this gives m (f,I) = m f¯, m = const, ε(m ) = ε +1.
2 2 2 2 2 M2
So, we have proved
Proposition 10
M (z|f,I) = {t′(z),f(z)}, m (f,I) = m f¯,
2 2 2
t′(z) ∈ E2k+1, ε(t′(z)) = ε , ε(m ) = ε +1.
n+ M2 2 M2
Further, it follows from (16) that dadM′(z|f,g,h) = 0, where M′(z|f,g) = M (z|f,g)−
2 2 2 2
f¯g¯t′(z).
So, the proposition follows from Theorem 3 and the condition M (z|f,g) ∈ D2k+1
2 n+
Proposition 11
M (z|f,g) = c m (z|f,g)+t (z)m (z|f,g)+c m (z|f,g)+m (z|f,g)+
2 1 2|1 D 2|2 3 2|3 2|ζ
+ ckm (z|f,g)+dadϕ (z|f,g),
5 2|5 1 D
where
←− 3
m (z|f,g) = f(z) ∂ ωAB∂ g(z), m¯ (|f,g) = 0, ε = 0,
2|1 A B 2|1 m
2|1
m (z|f,g) = f¯g¯, (cid:16) (cid:17)
2|2
m (z|f,g) = f¯E g(z)−(−1)ε(f)ε(g)g¯E f(z), m¯ (|f,g) = 0, ε = n ,
2|3 z z 2|3 m −
2|3
t (z) = t′(z)+c ∈ D2k+1, ε(t (z)) = ε ,
D 2 n+ D M2
m (z|f,g) = {ζ(z),g(z)}f¯−(−1)ε(f)ε(g){ζ(z),f(z)}g¯, m¯ (|f,g) = 0,
2|ζ 2|ζ
ζ(z) ∈ E2k+1/D2k+1, ε(ζ(z)) = ε +n ,
n+ n+ M2 −
m (z|f,g) = δ L2k+1, ε = 1, L2k+1 = 0 for k > 1,
2|5 n+,2 2 m2|5 2
ϕ (z|f) ∈ C (D2k+1,D2k+1), ε = ε ,
D 1 n+ n+ ϕD M2
M¯ (|f,g) = t¯ f¯g¯+cka ω(f,g)−ζ¯(|{f,g}),
2 D 5 k
20, k = 0,
3
ω(f,g) = δ dxdηdξf(x,ξ)g(x,η), a = 6, k = 1,
n+,2 k 
Z  0, k ≥ 2,
and the bilinear forms Li are defined by (11). 
2
To find m (f,g) and the relations between the parameters in Proposition 11 , consider
2
Eq. (16). It gives
dtrm′(f,g,h) = 3(m −t¯ )f¯g¯h¯ −cka [ω(f,g)h¯ +ω(f,h)g¯+ω(g,h)f¯], (28)
2 2 2 D 5 n−
where
m′(f,g) = m (f,g)−[ϕ¯ (|f)g¯−(−1)ε(f)ε(g)ϕ¯ (|g)f¯].
2 2 D D
10
Evidently, thereexistsuchfunctionsf,g,h ∈ D2k+1 thatf¯g¯h¯ 6= 0,{f,g} = {g,h} = {h,f} =
n+
0 and ω(f,g) = ω(g,h) = ω(h,f) = 0. Indeed, let
supp(f) supp(g) = supp(g) supp(h) = supp(h) supp(f) = ∅.
\ \ \
So, Eq. (28) implies
Proposition 12 m = t¯ ,
2 D
and as a consequence
dtrm′(f,g,h) = −cka [ω(f,g)h¯ +ω(f,h)g¯+ω(g,h)f¯] (29)
2 2 5 k
Proposition 13 ck = 0 for k ≥ 1, i.e. ck = c δ , where c ∈ K
5 5 5 0,k 5
Proof. Consider Eq. (29) for the functions f = g = h = ϕ(x)δ(ξ)
Proposition 14 Let k = 0. Then
3
ω(f,g)h¯ +ω(f,h)g¯+ω(g,h)f¯= dtrχ (f,g,h), (30)
20 2 2
where
20
¯
χ (f,g) = δ dzξf(z)g¯− dzξg(z)f . (31)
2 3 n+,2
(cid:18)Z Z (cid:19)
Proof.
¯ ¯
ω(f,g)h+ω(f,h)g¯+ω(g,h)f = δ dxf (x)g (x) dyh (y)+
n+,2 n− n− n−
(cid:16)Z Z
+ dxf (x)h (x) dyg (y)+ dxg (x)h (x) dyf (y) =
n− n− n− n− n− n−
Z Z Z Z (cid:17)
¯
= δ dzξ{f(z),g(z)}h+cycle(f,g,h)
n+,2
(cid:18)Z (cid:19)
Thus, we obtain
M (z|f,g) = c m (z|f,g)+c m (z|f,g)+m (z|f,g)+c δ δ L1(z|f,g)+
2 1 2|1 3 2|3 2|ζ 5 k,0 2,n+ 2
+t (z)m (z|f,g)+dadϕ(z|f,g),
D 2|2 1
m (f,g) = ϕ¯ (|f)g¯−(−1)ε(f)ε(g)ϕ¯ (|g)f¯+c δ χ (f,g)+b f¯g¯+µ ({f,g}),
2 D D 5 k,0 2 2 1
M (z|f,I) = {t (z),f(z)}, m (f,I) = t¯f¯.
2 D 2
where µ ∈ C (D2k+1,K), t ∈ D2k+1, ϕ ∈ C (D2k+1,D2k+1).
1 1 n+ D n+ D 1 n+ n+
Let M (f) = ϕ (z|f)−µ (f)I, M (I) = −t (z)−b I. Then
1D D 1 1D D 2
M (f,g) = M (f,g)+dadM (f,g) (32)
2 2|1 1 1D